Numerical Simulation of Forward and Inverse Problems of Internal Sound Field Based on Physics-informed Neural Network

doi:10.19596/j.cnki.1001-246x.8520

Abstract

Abstract:

A physics-informed neural network (PINN) is proposed for numerical simulation of forward and inverse problems associated with internal sound field in frequency domain. Unlike data-driven neural network, Helmholtz equation of a acoustic problem and corresponding boundary conditions are embedded in the neural network. The developed neural network reflects the distribution law of training data samples, and follows the physical law described by partial differential equations as well. For frequency acoustic problem with complex numbers, two types of networks are established. Verification and comparison are performed. Tedious numerical calculation processes such as meshing and numerical integration are not needed, and irregular domain and non-uniformly distributed nodes are freely addressed. Numerical examples, including the forward and inverse problems in two-dimensional and three-dimensional complex geometric structures, are provided to investigate effectiveness of the method. It shows that the PINN has good accuracy, convergence and robustness.

Key words: physics-informed neural network, acoustic problems, Helmholtz equation, forward problems, inverse problems

Guozheng WU, Fajie WANG, Suifu CHENG, Chengxin ZHANG. Numerical Simulation of Forward and Inverse Problems of Internal Sound Field Based on Physics-informed Neural Network[J]. Chinese Journal of Computational Physics, 2022, 39(6): 687-698.

Figures/Tables 18

Fig.1 A schematic of forward and inverse problems associated with internal sound field (a) forward problem; (b) inverse problem

Fig.2 A sketch of the first physics-informed neural network (PINN1)

Fig.3 A sketch of the second physics-informed neural network (PINN2)

Fig.4 Pipe acoustic model

Fig.5 Accurate and predicted sound pressures in a rectangular acoustic cavity

Fig.6 Absolute error distributions of two neural networks

Fig.7 Distributions of sound pressure at the lower boundary

Table 1 Numerical errors of PINN1, PINN2, LMFS and GFDM under same nodal distribution

方法	最大绝对误差	最大相对误差	均方根误差
PINN1	3.000 9 × 10^-5	1.692 3 × 10^-5	6.874 2 × 10^-7
PINN2	8.163 5 × 10^-4	7.302 9 × 10^-3	9.252 3 × 10^-5
LMFS	7.753 1 × 10^-6	4.183 7 × 10^-5	5.004 9 × 10^-6
GFDM	5.859 3 × 10^-2	1.593 6 × 10^-1	3.631 3 × 10^-2

Fig.8 Geometry model of a inverse problem with multiple connected domain

Fig.9 Calculation results of the two loss functions (a) calculation results; (b) absolute error

Fig.10 Contours of absolute error of two neural networks (a) PINN1; (b) PINN2

Fig.11 Exact results and predicted values with PINN1 on three unmeasurable boundaries

Fig.12 Exact results and predicted values with PINN2 on three unmeasurable boundaries

Fig.13 Distributions of absolute error of two networks under different noise levels

Fig.14 RMSEs of PINN1 and PINN2 with increasing noisy level

Fig.15 Three-dimensional acoustic cavity structures

Fig.16 Absolute errors of predicted values at internal nodes in three models

Table 2 Maximum relative errors and root mean square errors of PINN1 in acoustic analysis for three cavity structures

	最大相对误差	均方根误差
区域(a)	2.609 7×10^-5	4.009 5×10^-8
区域(b)	1.114 1×10^-4	2.989 6×10^-7
区域(c)	4.537 6×10^-4	1.149 2×10^-6

References 29

1	ESLAMI S M A , REZENDE D J , BESSE F , et al. Neural scene representation and rendering[J]. Science, 2018, 360 (6394): 1204- 1210. DOI
2	李野, 陈松灿. 基于物理信息的神经网络: 最新进展与展望[J]. 计算机科学, 2022, 49 (4): 254- 262.
3	ZHANG Qingfu , YAO Jun , HUANG Zhaoqin , et al. A multiscale deep learning model for fractured porous media[J]. Chinese Journal of Computational Physics, 2019, 36 (6): 665- 672.
4	BENOS L , TAGARAKIS A C , DOLIAS G , et al. Machine learning in agriculture: A comprehensive updated review[J]. Sensors, 2021, 21 (11): 3758. DOI
5	尧少波, 何伟峰, 陈丽华, 等. 融合物理的神经网络方法在流场重建中的应用[J]. 空气动力学学报, 2021, 39 (s): 1- 9.
6	HAN J , JENTZEN A , WEINAN E . Solving high-dimensional partial differential equations using deep learning[J]. Proceedings of the National Academy of Sciences, 2018, 115 (34): 8505- 8510. DOI
7	RAISSI M , PERDIKARIS P , KARNIADAKIS G E . Machine learning of linear differential equations using Gaussian processes[J]. Journal of Computational Physics, 2017, 348, 683- 693. DOI
8	王江, 陈文. 基于组合神经网络的时间分数阶扩散方程计算方法[J]. 应用数学和力学, 2019, 40 (7): 741- 750.
9	AARTS L P , VAN DER VEER P . Neural network method for solving partial differential equations[J]. Neural Processing Letters, 2001, 14 (3): 261- 271. DOI
10	JAGTAP A D , KHARAZMI E , KARNIADAKIS G E . Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 365, 113028. DOI
11	KRIZHEVSHY A , SUTSKEVER I , HINTON G E . Imagenet classification with deep convolutional neural networks[J]. Advances in Neural Information Processing Systems, 2012, 25, 1097- 1105.
12	PANG Guofei , LU Lu , KARNIADAKIS G E . fPINNs: Fractional physics-informed neural networks[J]. SIAM Journal on Scientific Computing, 2019, 41 (4): A2603- A2626. DOI
13	PANG G , D'ELIA M , PARKS M , et al. nPINNs: Nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator: Algorithms and applications[J]. Journal of Computational Physics, 2020, 422, 109760. DOI
14	RAISSI M , PERDIKARIS P , KARNIADAKIS G E . Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[J]. Journal of Computational Physics, 2019, 378, 686- 707. DOI
15	赵暾, 周宇, 程艳青, 等. 基于内嵌物理机理神经网络的热传导方程的正问题及逆问题求解[J]. 空气动力学学报, 2021, 39 (5): 19- 26.
16	JAGTAP A D , KHARAZMI E , KARNIADAKIS G E . Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 365, 113028. DOI
17	LU Lu , MENG Xuhui , MAO Zhiping , et al. DeepXDE: A deep learning library for solving differential equations[J]. SIAM Review, 2021, 63 (1): 208- 228. DOI
18	WANG Chao , WANG Fajie , GU Yan , et al. Simulation analysis of electrostatic field based on localized method of fundamental solutions[J]. Chinese Journal of Computational Physics, 2021, 38 (5): 612- 622.
19	王朝阳. 无网格广义有限差分法在Helmholtz正反问题中的应用[D]. 青岛: 青岛大学, 2018.
20	邓绍更, 闻建龙, 王军锋, 等. 流体力学反问题的类型及其应用[J]. 排灌机械, 2001, 19 (4): 41- 43.
21	HUA Qingsong , GU Yan , QU Wenzhen , et al. A meshless generalized finite difference method for inverse Cauchy problems associated with three-dimensional inhomogeneous Helmholtz-type equations[J]. Engineering Analysis with Boundary Elements, 2017, 82, 162- 171. DOI
22	NAIR V, HINTON G E. Rectified linear units improve restricted boltzmann machines[C]//Proceedings of the 27th International Conference on Internional Conference on Machine Learning, June 21-24, 2010: 807-814.
23	ZHENG Shuyu , PENG Jiazhen , ZHANG Xianmei , et al. Prediction of energy confinement time in Tokamak based on neural networks[J]. Chinese Journal of Computational Physics, 2021, 38 (4): 423- 430.
24	WANG Jiang , LIANG Yingjie , QIU Lin , et al. Improved machine learning technique for solving Hausdorff derivative diffusion equations[J]. Fractals, 2020, 28 (4): 2050071.
25	陆至彬, 瞿景辉, 刘桦, 等. 基于物理信息神经网络的传热过程物理场代理模型的构建[J]. 化工学报, 2021, 72 (3): 1496- 1503.
26	韦增欣, 袁功林, 连志钢. 解非线性对称方程组问题的近似高斯-牛顿基础BFGS方法[J]. 广西科学, 2004, 11 (2): 91- 99.
27	韩继业, 刘光辉. 无约束最优化线搜索一般模型及BFGS方法的整体收敛性[J]. 应用数学学报, 1995, (1): 112- 122.
28	陈增涛, 王发杰, 王超. 广义有限差分法在含阻抗边界空腔声学分析中的应用[J]. 力学学报, 2021, 53 (4): 1183- 1195.
29	WANG Fajie , FAN Chia-Ming , HUA Qingsong , et al. Localized MFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations[J]. Applied Mathematics and Computation, 2020, 364, 124658.

[1]	ZHOU Zhichao, XU Lingfei, REN Tianrong, GU Cunfeng. Aero-optical Effects in Supersonic Compression Ramp Flow Fields:GCV-FFT Method [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2020, 37(3): 284-298.
[2]	ZHOU Huanlin, YAN Jun, YU Bo, CHEN Haolong. Identification of Thermal Conductivity for Transient Heat Conduction Problems by Improved Cuckoo Search Algorithm [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2018, 35(2): 212-220.
[3]	WANG Qishen, LIU Minghui, ZHANG Lihua, HE Min. Reconstruction Flexural Stiffness of Symmetric Simple Supported Beams with Single Mode [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2014, 31(2): 216-222.
[4]	HE Wei, LI Qian, XU Zheng, ZHU Jinhua, HE Yangguang, WANG Lei. Analytical Solution of Forward Problem for Magnetic Induction Tomography in a Multi-layer Sphere Brain Model [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2010, 27(6): 912-918.
[5]	PAN Wenfeng, ZONG Zhixiong, YOU Yunxiang, MIAO Guoping. Fast Acoustic Far-field Inverse of Three-dimensional Scattering Objects in an Ocean Waveguide [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2010, 27(1): 121-130.
[6]	XUE Qi-wen, YANG Hai-tian, HU Guo-jun. A Conjugate Gradient Method for Inverse Heat Conduction Problems with Multi-variables in Transient-state [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2005, 22(1): 56-60.
[7]	Wu Jiming, Yu Dehao. The natural integral equation of 3-d exterior helmholtz problem and its numerical solution [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 1999, 16(5): 449-456.
[8]	Pan kiaosu. THE FINITE DIMENSIONAL NUMERICAL FORMULAR FOR INTERFACE ESTIMATION PROBLEM [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 1994, 11(4): 515-521.
[9]	Pan Xiaosu. THE ESTIMATION TECHNIQUE ABOUT THE INTERFACE OF POTENTIAL SYSTEMS [J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 1993, 10(4): 444-450.